Question: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+6y &= 7 \\ -5x+3y &= 5\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}-2x-6y &= -7\\ -10x+6y &= 10\end{align*}$ Add the top and bottom equations. $-12x = 3$ Divide both sides by $-12$ and reduce as necessary. $x = -\dfrac{1}{4}$ Substitute $-\dfrac{1}{4}$ for $x$ in the top equation. $2( -\dfrac{1}{4})+6y = 7$ $-\dfrac{1}{2}+6y = 7$ $6y = \dfrac{15}{2}$ $y = \dfrac{5}{4}$ The solution is $\enspace x = -\dfrac{1}{4}, \enspace y = \dfrac{5}{4}$.